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## Asymptotic results for multiplexing subexponential on-off processes (1998)

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Venue: | Advances in Applied Probability |

Citations: | 78 - 18 self |

### Citations

2210 | On the self-similar nature of ethernet traffic
- Leland, Taqqu, et al.
- 1993
(Show Context)
Citation Context ...ysis has increasingly shown that the traffic streams in modern broad-band networks exhibit long-tailed (subexponential) characteristics. For the case of Ethernet traffic such results were examined in =-=[LTW93]-=-. These statistical results have stimulated research in queueing analysis under the heavy-tailed (non Cramér) assumptions. Queueing analysis with self-similar long-range dependent arrival processes ap... |

1100 |
Regular variation
- Bingham, Goldie, et al.
- 1987
(Show Context)
Citation Context ...given by F(x) = 1− l(x) α ≥ 0, xα where l(x) : R+ → R+ is a function of slow variation, i.e., limx→∞l(δx)/l(x) = 1,δ > 1. These functions were invented by Karamata [KAR30] (the main reference book is =-=[BGT87]-=-). TheotherexamplesincludelognormalandsomeWeibulldistributions(see[KLU88,JLA95g]). 3. Analysis of the Aggregate Arrival Process This section consists of two parts. The first part is contained in Secti... |

901 |
Applied Probability and Queues
- Asmussen
- 1987
(Show Context)
Citation Context ..., beginning of the last activity period until time zero, i.e., Dc(0) = ∫ 0 t b 0 where tb 0 represents the beginning of the activity period that is still active at t = 0. Now, by Theorem 4.3, pp. 64, =-=[ASM87]-=-, it follows that the process {Tn + τon n ,−∞ < n < ∞} is also a stationary Poisson process with the same rate Λ. Therefore, the process A ∞,s t is reversible. This implies that Dc(0) is equal in dist... |

747 | Branching Processes,
- Athreya, Ney
- 1972
(Show Context)
Citation Context ...stributions In what follows we will state a few important results from the literature on subexponential distributions. The general relation between S and L is the following. Lemma 3 (Athreya and Ney, =-=[ATN72]-=-) S ⊂ L. Lemma 4 If F ∈ L then (1−F(x))e αx → ∞ as x → ∞, for all α > 0. Note: Lemma 4 clearly shows that for long-tailed distributions Cramér type conditions are not satisfied. The proof of the follo... |

433 |
Stochastic theory of data handling system with multiple sources
- Anick, Mitra, et al.
- 1982
(Show Context)
Citation Context ...ocess. Hence, investigating computationally tractable exact and approximate solution techniques are needed. For Markovian (fluid) On-Off processes a thorough investigation of this problem was done in =-=[AMS82]-=-. Many other results for multiplexing Markovian OnOff processes followed. These led to the Equivalent Bandwidth theory for Markovian, or, in general, exponentially bounded arrival processes; extensive... |

391 |
Introduction to Stochastic Processes
- Cinlar
- 1975
(Show Context)
Citation Context ...er r, any M/G/∞ process is uniquely defined with Possion arrival times {Tn}, and the lengths of On periods {τon n }; likewise, the pair {Tn}, {τon n } uniquely defines a compound Poisson process (see =-=[CIN75]-=-, pp. 90), that is a piecewise constant process with Poisson jump times {Tn}, and jump sizes {τon n }. Hence, proving (4.22) is equivalent to 18 Z d = Z y +Z y , (4.23) where Z,Z y ,Z y , are the comp... |

372 | A storage model with self-similar input. - Norros - 1994 |

297 |
The stability of a queue with non-independent interarrival and service times
- Loynes
- 1962
(Show Context)
Citation Context ... be a sequence of i.i.d. random variables that are driving a queueing process (Lindley’s recursion) Qn+1 = (Qn +Xn) + , n ≥ 0, (4.1) where q + = max(0,q). According to the classical result of Loynes’ =-=[LOY62]-=- under the stability condition EXn < 0 this recursion admits a unique stationary solution, and for all initial conditions P[Qn ≤ x] converges to the stationary distribution P[Q ≤ x]. For the rest of t... |

209 | Large deviations and overflow probabilities for the general single server queue, with application - Duffield, O’Connell - 1995 |

180 | Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. - Glynn, Whitt - 1994 |

160 | Large Deviations for Performance Analysis
- Shwartz, Weiss
- 1995
(Show Context)
Citation Context ...e obtained up to now apply for IR. In addition, directly from the definition it can be shown that F ∈ IR, ∫ ∞ ¯F(t)dt < ∞, ⇒ F1 ∈ IR. 0 Under the general large deviation Gärtner-Ellis conditions (see =-=[WES95]-=-) on the arrival process, it can be proved that the queue length distribution is exponentially bounded. To avoid statingGärtner-Ellisconditions, wewill define anarrival processet tobeexponential, if w... |

130 | Fundamental bounds and approximations for ATM multiplexers with applications to video teleconferencing,” - Elwalid, Heyman, et al. - 1995 |

120 |
Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences.
- Baccelli, Bremaud
- 2003
(Show Context)
Citation Context ... + = max(q,0). From the recursion above, it is clear that the process Qt is essentially the same as the G/G/1 workload process. Hence, by the fundamental stability theorem of Loynes (see Chapter 2 in =-=[BAB94]-=-) there exists a unique stationary process {Qs t ,−∞ < t < ∞} (P[Qs t ≤ x] = P[Q ≤ x]) that satisfies (4.4) (or equivalently (4.2)). In the rest of thepaper, whenever we refer to Qt, we will actually... |

113 | Source models for VBR broadcast video traffic. - Heyman, Lakshman - 1996 |

105 |
On the tails of waiting-time distributions.
- Pakes
- 1975
(Show Context)
Citation Context ...ractable approach using fluid renewal type models in which renewal times are long-tailed has been explored in [ANA95, HRS96]. Queueing results in these two papers rely onthe classical result by Pakes =-=[PAK75]-=- onthe subexponential (long-tailed) asymptotics of the waiting time distribution in a GI/GI/1 queue or in earlier work of Cohen [COH73] which considered a regularly varying GI/GI/1 queue. TheresultofP... |

102 | Analysis of an ATM buffer with self-similar (“fractal”) input traffic. - Likhanov, Tsybakov, et al. - 1995 |

89 | Subexponential distributions and integrated tails - Klüppelberg - 1988 |

88 |
Heavy tails and long range dependence in on/off processes and associated fluid models.
- Heath, Resnick, et al.
- 1998
(Show Context)
Citation Context ...dual process may be truly innocuous (like an On-Off process). The complex autocorrelation structure of the aggregate process obtained by multiplexing long-tailed On-Off processes has been examined in =-=[HRS96]-=-. General bounds for multiplexing long-tailed fluid processes have been derived in [CHW95]. In [BOX96] a limiting case of an infinite number of OnOff processes with regularly varying On distribution h... |

78 |
A theorem on sums of independent positive random variables and its applications to branching process,
- Chistyakov
- 1964
(Show Context)
Citation Context ...x→∞ ∗2 (x) = 2, (2.2) 1−F(x) where F∗2 denotes the 2-nd convolution of F with itself, i.e., F∗2 (x) = ∫ [0,∞) F(x−y)F(dy). The class of subexponential distributions was first introduced by Chistyakov =-=[CHI64]-=-. The definition is motivated by the simplification of the asymptotic analysis of convolution tails. One of the best known examples of distribution functions in S (and L) are functions of Regular Vari... |

74 | Waiting-time tail probabilities in queues with long-tail servicetime distributions. - Abate, Choudhury, et al. - 1994 |

66 |
Some results on regular variation for distributions in queueing and fluctuation theory.
- Cohen
- 1973
(Show Context)
Citation Context ...results in these two papers rely onthe classical result by Pakes [PAK75] onthe subexponential (long-tailed) asymptotics of the waiting time distribution in a GI/GI/1 queue or in earlier work of Cohen =-=[COH73]-=- which considered a regularly varying GI/GI/1 queue. TheresultofPakeshasbeengeneralizedtoaMarkovmodulatedsetting[AHK94,JLA95g]. In [AHK94] the subexponential asymptotics of a Markov modulated M/G/1 qu... |

65 | The Effect of Multiple Time Scales and Subexpo- nentialitiy in MPEG Video Streams on Queueing Behavior - Jelenkovic, Lazar, et al. - 1997 |

58 | Point process approaches to the modeling and analysis of self-similar traffic: Part I - Model construction. - Ryu, Lowen - 1996 |

56 | Subexponential asymptotics of a Markov-modulated random walk with queueing applications,”
- Jelenkovic, Lazar
- 1998
(Show Context)
Citation Context ...varying GI/GI/1 queue. TheresultofPakeshasbeengeneralizedtoaMarkovmodulatedsetting[AHK94,JLA95g]. In [AHK94] the subexponential asymptotics of a Markov modulated M/G/1 queue was investigated. Work in =-=[JLA95g]-=- further generalized these results to Markov modulated G/G/1 queues. In the same paper it was shown that a subexponential GI/GI/1 queue is invariantunder Markov modulation. In other words, a subexpon... |

54 | Convolution Tails, Product Tails and Domains of Attraction, Probability Theory and Related Fields - Cline - 1986 |

52 |
Asymptotic properties of supercritical branching processes
- Bingham, Doney
- 1975
(Show Context)
Citation Context ... theorem relates the tail behavior of a distribution function to the asymptotic behavior of its Laplace transform at the origin. For convenience we state the following result due to Bingham and Doney =-=[BID74]-=- ([BGT87], pp. 333). Let F beadistribution functionon[0,∞), andlet ˜ F(s)beitsLaplace-Stieltjes transform. Denotebymn = EX n = ∫ [0,∞) xn dF(x),n = 0,1,.... Whenmn < ∞, ˜ F(s)maybeexpanded26 in a Tay... |

51 |
Sur un mode de croissance régulière des fonctions
- Karamata
- 1930
(Show Context)
Citation Context ...ar Pareto family); F ∈ R−α if it is given by F(x) = 1− l(x) α ≥ 0, xα where l(x) : R+ → R+ is a function of slow variation, i.e., limx→∞l(δx)/l(x) = 1,δ > 1. These functions were invented by Karamata =-=[KAR30]-=- (the main reference book is [BGT87]). TheotherexamplesincludelognormalandsomeWeibulldistributions(see[KLU88,JLA95g]). 3. Analysis of the Aggregate Arrival Process This section consists of two parts. ... |

47 | Tail probabilities for non-standard risk and queueing processes with subexponential jumps. - Asmussen, Schmidli, et al. - 1999 |

44 |
Subexponential Distributions and Characterizations of Related Classes.” Probability Theory Related Fields
- KLUPPELBERG
- 1989
(Show Context)
Citation Context ...λEI N,on ˜ FN,1(s) . ˜FN,1(s) = π N 0 ˜ Ftr,N(s) 1−(1−π N 0 ) ˜ Ftr,N(s) ,7 or, in the time domain ¯FN,1(t) = π N 0 ∞∑ n=1 (1−π N 0 )n−1 F ∗n tr,N (t). (3.8) Now, F ∈ Sd implies F1 ∈ S (Theorem 1.1, =-=[KLU89]-=-), which by (3.7), and Lemma 5 (ii) (A), yields Ftr,N ∈ S. Hence, by Theorem 14 (A) and (3.8) it follows that ¯FN,1(t) ∼ π −N 0 ¯Ftr,N(t) ∼ Nπ1 1−π N 0 ¯F1(t) as t → ∞, (3.9) where the second asymptot... |

37 |
Subexponentiality and infinite divisibility Z.
- Embrechts, Goldie, et al.
- 1979
(Show Context)
Citation Context ...all α > 0. Note: Lemma 4 clearly shows that for long-tailed distributions Cramér type conditions are not satisfied. The proof of the following result can be found in Embrechts, Goldie and Veraverbeke =-=[EGV79]-=-. Lemma 5 Let F ∈ S. Then, (i) If G is a probability distribution such that ¯ G(x) = o( ¯ F(x)) as x → ∞, then F ∗G(x) ∼ ¯F(x) as x → ∞. (ii) If limx→∞ ¯ G(x)/ ¯ F(x) = c ∈ (0,∞), where G is a distrib... |

36 | On the superposition of renewal processes. - Cox, Smith - 1954 |

31 | A storage model with a two-state random environment,
- Kella, Whitt
- 1992
(Show Context)
Citation Context ...ss with On arrival rate r. In this subsection we assume that Off periods are also general (not necessarily exponential). (A general storage model in a two state random environment was investigated in =-=[KEW92]-=-.) Then, if we observe the queue at the beginning of On periods, the queue length Q P n evolves as follows (P stands for Palm probability [BAB94]). 13 Q P n+1 = (QP n +(r −c)τon n −cτoff n ) + , n ≥ 0... |

30 |
Superimposed renewal processes and storage with gradual input
- Cohen
- 1974
(Show Context)
Citation Context ...ocesses (sources); the main QOS parameter (performance measure) for this queueing system is the buffer occupancy probability distribution. The problem of multiplexing dates back to [RUB73, COH74]. In =-=[COH74]-=-, Cohen obtained a complete Laplace transform solution to this problem! (More recently, he revisited this problem in [COH94].) However, inverting the Laplace transform is usually a very tedious proces... |

29 |
Large claims approximations for risk processes in a Markovian environment
- Asmussen, Henriksen, et al.
- 1994
(Show Context)
Citation Context ...e distribution in a GI/GI/1 queue or in earlier work of Cohen [COH73] which considered a regularly varying GI/GI/1 queue. TheresultofPakeshasbeengeneralizedtoaMarkovmodulatedsetting[AHK94,JLA95g]. In =-=[AHK94]-=- the subexponential asymptotics of a Markov modulated M/G/1 queue was investigated. Work in [JLA95g] further generalized these results to Markov modulated G/G/1 queues. In the same paper it was shown ... |

28 | Asymptotic expansion for waiting time probabilities in an M=G=1 queue with long-tailed service time. Queueing Syst - Willekens, Teugels - 1992 |

26 | Long-tail Buffer-content Distributions in Broadband Networks.
- Choudhury, Whitt
- 1997
(Show Context)
Citation Context ...structure of the aggregate process obtained by multiplexing long-tailed On-Off processes has been examined in [HRS96]. General bounds for multiplexing long-tailed fluid processes have been derived in =-=[CHW95]-=-. In [BOX96] a limiting case of an infinite number of OnOff processes with regularly varying On distribution has been investigated. In the same paper (see also [BOX97]) a case of two processes, one of... |

26 | Superposition of point processes. - Çinlar - 1972 |

25 | Multiplexing on-off sources with sub exponential on periods: part1,” in - Jelenkovic, Lazar - 1997 |

24 | Fluid queues and regular variation
- Boxma
- 1996
(Show Context)
Citation Context ... the aggregate process obtained by multiplexing long-tailed On-Off processes has been examined in [HRS96]. General bounds for multiplexing long-tailed fluid processes have been derived in [CHW95]. In =-=[BOX96]-=- a limiting case of an infinite number of OnOff processes with regularly varying On distribution has been investigated. In the same paper (see also [BOX97]) a case of two processes, one of which had r... |

23 | Multiple time scales and subexponentiality in MPEG video streams - Jelenković, Lazar, et al. - 1996 |

19 |
Intermediate regular and π variation
- Cline
(Show Context)
Citation Context ...bution function F is intermediate regular varying F ∈ IR if 15 limliminf δ↓1 t→∞ ¯F(δt) ¯F(t) = 1. Remark: For recent results on distributions of intermediate regular variation we refer the reader to =-=[CLI94]-=-. Some basic properties of IR are: IR ⊂ S; R ⊂ IR. Also, it is not very difficult to see that IR ⊂ Sd. Therefore, all of the results that we have obtained up to now apply for IR. In addition, directly... |

18 | Asymptotics of Palm-stationary buffer content distributionin fluid flow queues - Rolski, Schlegel, et al. - 1999 |

16 | The output of a buffered data communication system - Rubinovitch - 1973 |

13 | Regular variation in a multi-source fluid queue
- Boxma
- 1997
(Show Context)
Citation Context ...d processes have been derived in [CHW95]. In [BOX96] a limiting case of an infinite number of OnOff processes with regularly varying On distribution has been investigated. In the same paper (see also =-=[BOX97]-=-) a case of two processes, one of which had regularly varying On periods and the other had exponential On periods, has been solved. Similar scenario with intermediately regularly varying On periods ha... |

13 | Intermediate regular and variation. - Cline - 1994 |

12 | On the sojourn time of sessions at an ATM buffer with long-range dependent input traffic - Anantharam - 1995 |

11 | Georganas, "Analysis of an ATM buffer with self-similar ("fractal") input traffic - Likhanov, Tsybakov, et al. - 1995 |

6 |
On the effective bandwidth in buffer design for the multiserver channels
- Cohen
- 1994
(Show Context)
Citation Context ... distribution. The problem of multiplexing dates back to [RUB73, COH74]. In [COH74], Cohen obtained a complete Laplace transform solution to this problem! (More recently, he revisited this problem in =-=[COH94]-=-.) However, inverting the Laplace transform is usually a very tedious process. Hence, investigating computationally tractable exact and approximate solution techniques are needed. For Markovian (fluid... |

6 | Tail probabilitites for M/G/1 input processes (I): Preliminary asymptotices. Queueing Syst - Parulekar, Makowski - 1997 |

5 | Tail probabilitites for a multiplexer with self-similar traffic - Parulekar, Makowski - 1996 |

5 |
Performance decay in a single server queueing model with long range dependence
- Resnick, Samorodnitsky
- 1996
(Show Context)
Citation Context ...cesses, one of which had regularly varying On periods and the other had exponential On periods, has been solved. Similar scenario with intermediately regularly varying On periods has been examined in =-=[RSS97]-=-. The literature does not explicitly give precise asymptotic results for the case of multiplexing two or more long-tailed processes. From a mathematical point of view the new results in this paper wer... |

5 |
Tail probabilitites for M/G/ input processes (I): preliminary asymptotices
- Parulekar, Makowski
- 1997
(Show Context)
Citation Context ...r of customers in service in an M/G/∞ queue in which the customer service requirement has the same distribution as τon and the arrival rate is Λ. (For recent asymptotic results on M/G/∞ processes see =-=[PAM97]-=-.) Note that Theorem 3 implies that lim N→∞ lim t→∞ λN→Λ P[I N,on > t] P[τ on > t] = eΛτon. (3.14) However, this does not necessarily imply that we can interchange the limit and derive the asymptotics... |

4 |
Rare events in the presence of heavy tails
- Asmussen
- 1996
(Show Context)
Citation Context ...bexponential distributions (and further references) can be found in [CLI86, KLU88]. For a recent survey of the application of subexponential distributions in queueing theory the reader is referred to =-=[ASM96]-=-. B: Proof of Theorem 6 As we have already mentioned the proof of this result is based on Karamata’s Tauberian/Abelian theorem for distribution functions of regular variation. This theorem relates the... |

4 | Multiple time scale and subexponential asymptotic behavior of a network multiplexer - Jelenković, Lazar - 1996 |

3 | Teugels, "Asymptotic expansion for waiting time probabilities in an M/G/1 queue with long-tailed service time - Willekens, L - 1992 |

1 |
Tailprobabilitiesfornon-standardriskandqueueing processes with subexponential jumps.” preprint
- Asmussen, Schmidli, et al.
- 1997
(Show Context)
Citation Context ...e analysis of a subexponential semi-Markov fluid queue ([JLA95g]). Further generalizations of the result in [JLA95g] to arrival processes with a more complex dependency structure were investigated in =-=[ASC97]-=-. Asymptotic expansion refinements of Pakes’ result can be found in [WIT92, ACW94]. The analysis of a fluid queue in which more than one long-tailed process is multiplexed appearsto beavery difficult ... |

1 | Fundamental bounds and approximationsforATM multiplexerswithapplicationstovideoteleconferencing - Elwalid, Heyman, et al. - 1995 |